fractions

Business Math Study Guide 2 Page 1
FRACTION REVIEW
A. INTRODUCTION
1. What is a fraction?
A fraction consists of a numerator (part) on top of a
denominator (total) separated by a horizontal line.
For example, the fraction of the circle which is shaded is:
2 (parts shaded)
4 (total parts)
In the square on the right, the fraction shaded is
8
3
and
the fraction unshaded is
8
5
Fraction = numerator
denominator
2. Equivalent Fractions – Multiplying
The three circles on the right each have
equal parts shaded, yet are represented
by different but equal fractions. These
fractions, because they are equal, are
called equivalent fractions.
Any fraction can be changed into an equivalent fraction
by multiplying both the numerator and denominator by the same number
2
1
=
4
2
or
2
1
8
4
so
8
4
4
2
2
1
Similarly
9
5
=
18
10
or
9
5
=
27
15
so
9
5
=
18
10
=
27
15
You can see from the above examples that each fraction has an infinite number of fractions
that are equivalent to it.
8
4
4
2
2
1
x 2
x 2
x 3
x 3
x 2
x 2
x 4
x 4 =

Business Math Study Guide 2 – Fractions FB/2015 Page 2
3. Equivalent Fractions – Dividing (Reducing)
Equivalent fractions can also be created if both the
numerator and denominator can be divided by the
same number (a factor) evenly.
This process is called “reducing a fraction” by
dividing a common factor (a number which divides
into both the numerator and denominator evenly).
2
1
4
4
8
4
9
3
9
9
81
27 
6
1
5
5
30
5
5
3
2
2
10
6
4. Simplifying a Fraction (Reducing to its Lowest Terms)
It is usual to reduce a fraction until it can’t be reduced
any further.
A simplified fraction has no common factors which will
divide into both numerator and denominator.
Notice that, since 27 and 81 have a common factor of 9,
we find that
9
3
is an equivalent fraction.
But this fraction has a factor of 3 common to both
numerator and denominator.
So, we must reduce this fraction again. It is difficult to
see, but if we had known that 27 was a factor (divides
into both parts of the fraction evenly), we could have
arrived at the answer in one step
e.g.
3
1
8
8
24
8
4
3
15
15
60
45 
9
3
9
9
81
27 
3
1
3
3
9
3
3
1
27
27
81
27 

Business Math Study Guide 2 – Fractions FB/2015 Page 3
5. EXERCISE 1: Introduction to Fractions
a) Find the missing part of these equivalent fractions
Example:
5
3
=
10
Since 5 x 2 = 10,
multiply the numerator
by 2, also.
So,
5
3
=
10
6
1)
3
2
=
6
2)
4
3
=
12
3)
8
5
=
40
4)
16
1
=
32
5)
15
2
=
45
6.)
9
7
=
27
7)
10
7
=
100
8)
4
3
=
44
b) Find the missing part of these equivalent fractions.
Example:
10
5
=
2
Since 10 ÷ 5 = 2 divide
the numerator by 5, also.
So,
10
5
=
2
1
1)
16
8
=
4
2)
27
24
=
9
3)
10
6
=
5
4)
35
25
=
7
5)
30
20
=
6
6)
100
90
=
50
c) Simplify the following fractions (reduce to lowest terms).
1)
12
9
2)
12
8
3)
8
6
4)
20
15
5)
25
20
6)
21
14
7)
16
8
8)
36
24
9)
99
66
10)
30
18
x 2
x 2
÷ 5
÷ 5

Business Math Study Guide 2 – Fractions FB/2015 Page 4
B. TYPES OF FRACTIONS
1. Common Fractions
A common fraction is one in which the numerator is less than the denominator
(or a fraction which is less than the number 1). A common fraction can also be called a proper fraction.
e.g.
2
1
,
4
3
,
93
88
,
15
8
are all common fractions.
2. Fractions that are Whole Numbers
Some fractions, when reduced, are really whole numbers (1, 2, 3, 4… etc).
Whole numbers occur if the denominator divides into the numerator evenly.
e.g.
4
8
is the same as
4
8
÷
4
4
=
1
2
or 2
5
30
is the same as
5
30
÷
5
5
=
1
6
or 6
So, the fraction
5
30
is really the whole number 6.
Notice that a whole number can always be written as a fraction with a denominator of 1.
e.g. 10 =
1
10
3. Mixed Numbers
A mixed number is a combination of a whole number and a common fraction.
e.g. 2
5
3
(two and three-fifths)
27
9
2
(twenty-seven and two-ninths)
9
6
3
= 9
2
1
(always reduce fractions)

Business Math Study Guide 2 – Fractions FB/2015 Page 5
4. Improper Fractions
An improper fraction is one in which the numerator is larger than the
denominator.
From the circles on the right, we see that 1
4
3
(mixed number)
is the same as
4
7
(improper fraction).
An improper fraction, like
4
7
, can be changed to a mixed number by
dividing the denominator into the numerator and expressing the
remainder (3) as the numerator.
e.g.
5
16
= 3
5
1
8
29
= 3
8
5
3
14
= 4
3
2
A mixed number can be changed to an improper fraction by changing
the whole number to a fraction with the same denominator as the
common fraction.
4
7
= 4
1
7
= 1
4
3
34
2
5
3
=
5
10
and
5
3
=
5
13
10
9
1
=
9
90
and
9
1
=
9
91
A simple way to do this is to multiply the whole number by the denominator,
and then add the numerator.
e.g. 4
9
5
=
959x4 
=
9536 
=
9
41
10
7
2
=
727x10 
=
7270 
=
7
72
5. Simplifying fractions
All types of fractions must always be simplified (reduced to lowest terms).
e.g.
9
6
=
3
2
, 2
25
5
= 2
5
1
,
18
27
=
2
3
= 1
2
1
Note that many fractions can not be reduced since they have no common factors.
e.g.
21
17
,
9
4
,
37
18
1
4
3
=
4
7